{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plo t" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 32 "SISTEMAS NO LINEALES EN EL PLANO" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 10 "Objetivos:" }}{PARA 0 "" 0 "" {TEXT -1 64 "- Exten der la teor\355a de sistemas lineales a sistemas No Lineales" }}{PARA 0 "" 0 "" {TEXT -1 61 "- Calcular y clasificar todos los puntos cr\355 ticos en el plano" }}{PARA 0 "" 0 "" {TEXT -1 65 "- Construir diagrama s de fase de sistemas no lineales en el plano" }}{PARA 0 "" 0 "" {TEXT -1 68 "- Aplicar la teor\355a para resolver problemas de modela ci\363n sencillos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 12 "Introducci\363n" }}{PARA 0 "" 0 "" {TEXT -1 105 "Se ver\341 el m\351todo de linealizaci\363n y las definiciones de puntos cr\355ticos hiperb\363licos y no-hiperb\363licos. " }} {PARA 0 "" 0 "" {TEXT -1 63 "Diagramas de fases se construir\341n usan do el Teorema de Hartman." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT 261 0 "" }{TEXT 262 35 "Li nealizaci\363n y Teorema de Hartman." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Suponga que el sistema autonomo no line al:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "diff(x(t),t)=P(x(t),y(t)), diff(y(t),t)=Q(x(t),y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%%diff G6$-%\"xG6#%\"tGF*-%\"PG6$F'-%\"yGF)/-F%6$F.F*-%\"QGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "tiene un punto cr\355tico (u,v)." }}{PARA 0 "" 0 "" {TEXT -1 171 "Lo primero para linealizar entorno a ese punto cr\355tico ser\341 tomar una transformaci\363n lineal que mueva el punto cr\355tico al o r\355gen y aplicar la expanci\363n en serie de Taylor." }}{PARA 0 "" 0 "" {TEXT -1 65 "Sea X = x - u ; Y = y - v. Entonces el sistema qued ar\341 dado por:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "diff(X(t),t) = P(X+u,Y+v),diff(X(t),t)=mtaylo r(P(X+u,Y+v), [X=0,Y=0],2)+R(X,Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ /-%%diffG6$-%\"XG6#%\"tGF*-%\"PG6$,&F(\"\"\"%\"uGF/,&%\"YGF/%\"vGF//F$ ,*-F,6$F0F3F/*&--&%\"DG6#F/6#F,F7F/F(F/F/*&--&F<6#\"\"#F>F7F/F2F/F/-% \"RG6$F(F2F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "diff(Y(t),t ) = Q(X+u,Y+v),diff(Y(t),t)=mtaylor(Q(X+u,Y+v), [X=0,Y=0],2)+S(X,Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%%diffG6$-%\"YG6#%\"tGF*-%\"QG6$,& %\"XG\"\"\"%\"uGF0,&F(F0%\"vGF0/F$,*-F,6$F1F3F0*&--&%\"DG6#F06#F,F7F0F /F0F0*&--&F<6#\"\"#F>F7F0F(F0F0-%\"SG6$F/F(F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "luego de haber aplicado la expansi\363n en serie de T aylor. Los t\351rminos no lineales R y S satisfacen la condici\363n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Limit(R/r, r=0)=0;Limit(S/ r, r=0)=0; r=sqrt(X*X+Y*Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&Lim itG6$*&%\"RG\"\"\"%\"rG!\"\"/F*\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&%\"SG\"\"\"%\"rG!\"\"/F*\"\"!F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%\"rG*$-%%sqrtG6#,&*$)%\"XG\"\"#\"\"\"F.*$)%\"YGF-F. F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 "Las funciones R y S se d icen que son de orden r*r (O(r*r)). Descartando los t\351rminos no lin eales y notando y Q(u,v) y P(u,v) valen cero pues (u,v) es un punto cr \355tico, el sistema LINEALIZADO tendr\341 la forma:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "diff(X(t),t)=X(t)*(diff(P(x=u,y=v),x))+Y (t)*(diff(P(x=u,y=v),y));diff(Y(t),t)=X(t)*(diff(Q(x=u,y=v),x))+Y(t)*( diff(Q(x=u,y=v),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-% \"XG6#%\"tGF*,&*&F'\"\"\"-F%6$-%\"PG6$/%\"xG%\"uG/%\"yG%\"vGF4F-F-*&-% \"YGF)F--F%6$F0F7F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-% \"YG6#%\"tGF*,&*&-%\"XGF)\"\"\"-F%6$-%\"QG6$/%\"xG%\"uG/%\"yG%\"vGF6F/ F/*&F'F/F0F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "En su forma mat ricial:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "diff(x(t),t)=J(u,v)*x(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"xG6#%\"tGF**&-%\"JG6$%\"uG%\"vG\"\"\"F'F 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Donde J(u,v) ser\341 la matr iz Jacobiana evaluada en el punto (u,v)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 15 "Definici\363n 1.- \+ " }{TEXT -1 28 "Un punto cr\355tico se llamar\341 " }{TEXT 264 11 "hip erb\363lico" }{TEXT -1 203 " si la parte real de los valores propios d e la matriz Jacobiana J(u,v) no valen cero. Si la parte real de cualqu iera de los valores propios del Jacobiano valen cero, entonces el punt o cr\355tico se llamar\341 " }{TEXT 265 15 "no hiperb\363lico." } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 20 "TEOREMA DE HARTMAN.-" }}{PARA 0 "" 0 "" {TEXT -1 413 "Suponga que (u,v) es un punto cr\355tico hiperb\363lico del si stema no lineal. Entonces hay una vecindad en torno a este punto cr \355tico en el cual el diagrama de fase del sistema no lineal es el mi smo que el del sistema linealizado. En otras palabras, hay un cambio c ontinuo curvilineo entre un diagrama de fase y otro, y en una peque \361a regi\363n alrededor del punto cr\355tico los diagramas son cuali tativamente equivalentes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 280 "Noten que las curvas de estabilidad e inestabilid ad de el sistema no lineal ser\341n tangentes a las curvas de el siste ma linelizado cerca del punto cr\355tico relevante. Esas trayectorias \+ divergen a medida que uno se aleja del punto cr\355tico. Esto lo ilust raremos en los ejemplos 1 y 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 267 10 "Notaci\363n.-" }{TEXT -1 88 " Rectas estables y no estables de sistemas no lineales se notan Ws y Wu respectivamente." }}{PARA 0 "" 0 "" {TEXT -1 97 "El teorema de H artman implica que Ws y Wu son tangentes a Es y Eu en el punto critico relevante. " }}{PARA 0 "" 0 "" {TEXT -1 172 "Si cualquiera de los pu ntos cr\355ticos es NO HIPERBOLICO, entonces otros m\351todos son usad os para bosquejar un diagrama de fase, y soluciones num\351ricas puede n ser requeridas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 0 "" }{TEXT 269 0 "" }{TEXT 270 38 "CONSTRUYE NDO DIAGRAMAS DE FASE PLANOS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 110 "El m\351todo para bosquejar los diagrama s de fase teniendo puntos cr\355ticos hiperb\363licos se resumen en tr es pasos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "- Calcular todos los puntos cr\355ticos." }}{PARA 0 "" 0 "" {TEXT -1 136 "- Linealizar (calcular la matriz Jacobiana y evaluar en cada p unto cr\355tico) y clasificar cada punto cr\355tico seg\372n el Teorem a de Hartman." }}{PARA 0 "" 0 "" {TEXT -1 85 "- Determinar las isoclin as y usar dy/dx para obtener direcciones de las trayectorias." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 0 "" }{TEXT 272 10 "EJEMPLO 1." }}{PARA 0 "" 0 "" {TEXT -1 50 "Bos queje el diagrama de fase del sistema no lineal" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "diff(x(t),t)=x(t);diff(y(t),t)=x(t)*x(t)+y(t)* y(t)-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"xG6#%\"tGF*F '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"tGF*,(*$)-% \"xGF)\"\"#\"\"\"F1*$)F'F0F1F1F1!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 273 9 "Soluci\363n:" }}{PARA 0 "" 0 "" {TEXT -1 66 "Primero localizamos puntos cr\355ticos resolviendo dx/dt=0 y dy/dt =0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(\{x=0, x*x+y*y-1=0\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"xG\"\"!/%\"yG\"\"\"<$F$/F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Entonces hay dos puntos cr\355ticos (0,1) y (0, -1)." }}{PARA 0 "" 0 "" {TEXT -1 46 "Segundo es calcular el Jacobiano, y obtenemos:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 " Warning, the protected names norm and trace have been redefined and un protected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A := vector( [x, x*x+y*y-1] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "jacobian(A, [ x,y]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'vectorG6#7$%\"xG,( *$)F)\"\"#\"\"\"F.*$)%\"yGF-F.F.F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"\"\"!7$,$%\"xG\"\"#,$%\"yGF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "Lo tercero es evaluar el Jacobiano en ca da punto cr\355tico y determinar el tipo de punto cr\355tico que es." }}{PARA 0 "" 0 "" {TEXT -1 50 "Para el primer punto cr\355tico se obti ene la matriz:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "J(0,1):=m atrix(2,2,[1,0,0,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"JG6$\"\" !\"\"\"-%'matrixG6#7$7$F(F'7$F'\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "La matriz es diagonal, sus valores propios son 1 y 2, los cual es son reales positivos (igual signo) y distintos, por lo que tenemos \+ un " }{TEXT 274 15 "NODO INESTABLE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 51 "Para el segundo punto cr\355tico se obti ene la matriz:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "J(0,-1):= matrix(2,2,[1,0,0,-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"JG6$\" \"!!\"\"-%'matrixG6#7$7$\"\"\"F'7$F'!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Matriz diagonal, valores propios 1 y -2, reales de disti nto signo, por lo que el punto cr\355tico es tipo " }{TEXT 275 6 "SILL A." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "No ten que como las matrices son diagonales, sus vectores propios para am bos puntos criticos son (1,0) y (0,1)." }}{PARA 0 "" 0 "" {TEXT -1 343 "Entonces en las cercan\355as de cada punto cr\355tico, Ws y Wu so n tangentes a las lineas generadas por cada vector propio atrav\351s d el punto cr\355tico. Entonces cerca de cada punto cr\355tico las recta s son HORIZONTALES y VERTICALES. Noten que Ws y Wu no necesariamente s on rectas, lo que si son siempre tangentes a Es y Eu en el punto cr \355tico relevante." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 34 "Consideremos ahora las ISOCLINAS. " }}{PARA 0 "" 0 "" {TEXT -1 165 "dx/dt=0 cuando x=0, y en esta linea dy/dt=y*y-1. Entonce s si el modulo de y es menor que uno, entonces dy/dt<0 y si el modulo \+ de y es mayor que uno entonces dy/dt>0." }}{PARA 0 "" 0 "" {TEXT -1 138 "dy/dt=0 en el circulo de centro (0,0) y radio 1 (x*x+y*y=1), y en esta curva dx/dt=x. Entonces si x>0 dx/dt>0 y si x<0 entonces dx/dt< 0. " }}{PARA 0 "" 0 "" {TEXT -1 47 "Las trayectorias estar\341n dadas \+ por la ecuacion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "diff(y(x),x)=(x*x+y(x)*y(x)-1)/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"yG6#%\"xGF**&,(*$)F*\"\"#\"\" \"F0*$)F'F/F0F0F0!\"\"F0F*F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dsolve(%,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"x G,$*&,**&-%(BesselJG6$\"\"!F'\"\"\"F'F0F0-F-6$F0F'!\"\"*(%$_C1GF0-%(Be sselYGF.F0F'F0F0*&F5F0-F7F2F0F3F0,&F8F0F1F0F3F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Poniendo todo esto en el diagrama de fase se obtiene :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warnin g, the name adjoint has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sys:=diff(y(t),t)=x(t)*x(t)+y(t)*y(t)-1,diff(x(t),t)= x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG6$/-%%diffG6$-%\"yG6#%\"t GF-,(*$)-%\"xGF,\"\"#\"\"\"F4*$)F*F3F4F4F4!\"\"/-F(6$F1F-F2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "iniset:=seq(seq([0,i,j],i=-2 ..2),j=-2..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "DEplot( \{sys\},[x(t),y(t)],t=-3..3,\{iniset,[0,0.1,-1],[0,-0.1,-1]\},stepsize =0.1,x=-3..3,y=-3..3,color=black,linecolor=blue);" }}{PARA 7 "" 1 "" {TEXT -1 172 "Warning, x is present as both a dependent variable and a name. 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